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Mirrors > Home > MPE Home > Th. List > finlocfin | Structured version Visualization version GIF version |
Description: A finite cover of a topological space is a locally finite cover. (Contributed by Jeff Hankins, 21-Jan-2010.) |
Ref | Expression |
---|---|
finlocfin.1 | ⊢ 𝑋 = ∪ 𝐽 |
finlocfin.2 | ⊢ 𝑌 = ∪ 𝐴 |
Ref | Expression |
---|---|
finlocfin | ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ 𝑋 = 𝑌) → 𝐴 ∈ (LocFin‘𝐽)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1135 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ 𝑋 = 𝑌) → 𝐽 ∈ Top) | |
2 | simp3 1137 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ 𝑋 = 𝑌) → 𝑋 = 𝑌) | |
3 | simpl1 1190 | . . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ 𝑋 = 𝑌) ∧ 𝑥 ∈ 𝑋) → 𝐽 ∈ Top) | |
4 | finlocfin.1 | . . . . . 6 ⊢ 𝑋 = ∪ 𝐽 | |
5 | 4 | topopn 22928 | . . . . 5 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽) |
6 | 3, 5 | syl 17 | . . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ 𝑋 = 𝑌) ∧ 𝑥 ∈ 𝑋) → 𝑋 ∈ 𝐽) |
7 | simpr 484 | . . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ 𝑋 = 𝑌) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋) | |
8 | simpl2 1191 | . . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ 𝑋 = 𝑌) ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ Fin) | |
9 | ssrab2 4090 | . . . . 5 ⊢ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑋) ≠ ∅} ⊆ 𝐴 | |
10 | ssfi 9212 | . . . . 5 ⊢ ((𝐴 ∈ Fin ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑋) ≠ ∅} ⊆ 𝐴) → {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑋) ≠ ∅} ∈ Fin) | |
11 | 8, 9, 10 | sylancl 586 | . . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ 𝑋 = 𝑌) ∧ 𝑥 ∈ 𝑋) → {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑋) ≠ ∅} ∈ Fin) |
12 | eleq2 2828 | . . . . . 6 ⊢ (𝑛 = 𝑋 → (𝑥 ∈ 𝑛 ↔ 𝑥 ∈ 𝑋)) | |
13 | ineq2 4222 | . . . . . . . . 9 ⊢ (𝑛 = 𝑋 → (𝑠 ∩ 𝑛) = (𝑠 ∩ 𝑋)) | |
14 | 13 | neeq1d 2998 | . . . . . . . 8 ⊢ (𝑛 = 𝑋 → ((𝑠 ∩ 𝑛) ≠ ∅ ↔ (𝑠 ∩ 𝑋) ≠ ∅)) |
15 | 14 | rabbidv 3441 | . . . . . . 7 ⊢ (𝑛 = 𝑋 → {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} = {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑋) ≠ ∅}) |
16 | 15 | eleq1d 2824 | . . . . . 6 ⊢ (𝑛 = 𝑋 → ({𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin ↔ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑋) ≠ ∅} ∈ Fin)) |
17 | 12, 16 | anbi12d 632 | . . . . 5 ⊢ (𝑛 = 𝑋 → ((𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin) ↔ (𝑥 ∈ 𝑋 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑋) ≠ ∅} ∈ Fin))) |
18 | 17 | rspcev 3622 | . . . 4 ⊢ ((𝑋 ∈ 𝐽 ∧ (𝑥 ∈ 𝑋 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑋) ≠ ∅} ∈ Fin)) → ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin)) |
19 | 6, 7, 11, 18 | syl12anc 837 | . . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ 𝑋 = 𝑌) ∧ 𝑥 ∈ 𝑋) → ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin)) |
20 | 19 | ralrimiva 3144 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ 𝑋 = 𝑌) → ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin)) |
21 | finlocfin.2 | . . 3 ⊢ 𝑌 = ∪ 𝐴 | |
22 | 4, 21 | islocfin 23541 | . 2 ⊢ (𝐴 ∈ (LocFin‘𝐽) ↔ (𝐽 ∈ Top ∧ 𝑋 = 𝑌 ∧ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ {𝑠 ∈ 𝐴 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈ Fin))) |
23 | 1, 2, 20, 22 | syl3anbrc 1342 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ 𝑋 = 𝑌) → 𝐴 ∈ (LocFin‘𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∀wral 3059 ∃wrex 3068 {crab 3433 ∩ cin 3962 ⊆ wss 3963 ∅c0 4339 ∪ cuni 4912 ‘cfv 6563 Fincfn 8984 Topctop 22915 LocFinclocfin 23528 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-om 7888 df-1o 8505 df-en 8985 df-fin 8988 df-top 22916 df-locfin 23531 |
This theorem is referenced by: locfincmp 23550 cmppcmp 33819 |
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